Multiple periodic solutions to a class of second-order nonlinear mixed-type functional differential equations

where τ, σ , and c are constants in R with τ ≥ 0, σ ≥ 0, |c| < 1, g(t,x) is a T(> 0)-periodic function in t > 0, and for an arbitrary bounded domain E ⊂ R, g(t,x) is a Lipschitz function in [0,T] × E, p ∈ C(R,R), p(t +T) = p(t), and ∫ T 0 p(t)dt = 0. They obtained some sufficient conditions to guarantee the existence, at least a T-periodic solution, for this system. But, for the existence of periodic solutions of functional differential equations, one commonly uses methods of fixed point theory, coincidence degree theory, Fourier analysis, and so forth. Critical point theory has rarely been used. In [10, 11], the authors obtained multiple periodic solutions for a class retarded differential equations by means of critical point theory and Zp group index theory. Nevertheless, we noted that these results were obtained by reducing retarded differential equations to related ordinary differential equations. The purpose of our paper is to establish a kind of variational framework with delayed variables for a class of mixed-type differential equations. Unlike [10, 11], our approach enables us to obtain, by critical point theory and Z2 group index theory, the existence of nontrivial periodic solutions to such equations without reducing it to the one of ordinary differential equations. Subsequently, we introduce Z2 group index theory and knowledge about critical points.